Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{q^2 - q}{q^2 + 3q - 4}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - q}{q^2 + 3q - 4} = \dfrac{(q)(q - 1)}{(q + 4)(q - 1)} $ Notice that the term $(q - 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 1)$ gives: $x = \dfrac{q}{q + 4}$ Since we divided by $(q - 1)$, $q \neq 1$. $x = \dfrac{q}{q + 4}; \space q \neq 1$